NAME
CGEHRD - reduce a complex general matrix A to upper Hessen-
berg form H by a unitary similarity transformation
SYNOPSIS
SUBROUTINE CGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK,
INFO )
INTEGER IHI, ILO, INFO, LDA, LWORK, N
COMPLEX A( LDA, * ), TAU( * ), WORK( LWORK )
PURPOSE
CGEHRD reduces a complex general matrix A to upper Hessen-
berg form H by a unitary similarity transformation: Q' * A
* Q = H .
ARGUMENTS
N (input) INTEGER
The order of the matrix A. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER It is assumed that A is
already upper triangular in rows and columns 1:ILO-1
and IHI+1:N. ILO and IHI are normally set by a pre-
vious call to CGEBAL; otherwise they should be set
to 1 and N respectively. See Further Details. If N >
0,
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the N-by-N general matrix to be reduced.
On exit, the upper triangle and the first subdiago-
nal of A are overwritten with the upper Hessenberg
matrix H, and the elements below the first subdiago-
nal, with the array TAU, represent the unitary
matrix Q as a product of elementary reflectors. See
Further Details. LDA (input) INTEGER The lead-
ing dimension of the array A. LDA >= max(1,N).
TAU (output) COMPLEX array, dimension (N-1)
The scalar factors of the elementary reflectors (see
Further Details). Elements 1:ILO-1 and IHI:N-1 of
TAU are set to zero.
WORK (workspace) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input) INTEGER
The length of the array WORK. LWORK >= max(1,N).
For optimum performance LWORK >= N*NB, where NB is
the optimal blocksize.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal
value.
FURTHER DETAILS
The matrix Q is represented as a product of (ihi-ilo) ele-
mentary reflectors
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a complex scalar, and v is a complex vector
with v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi)
is stored on exit in A(i+2:ihi,i), and tau in TAU(i).
The contents of A are illustrated by the following example,
with n = 7, ilo = 2 and ihi = 6:
on entry on exit
( a a a a a a a ) ( a a h h h h
a ) ( a a a a a a ) ( a h h h
h a ) ( a a a a a a ) ( h h h
h h h ) ( a a a a a a ) ( v2 h
h h h h ) ( a a a a a a ) ( v2
v3 h h h h ) ( a a a a a a ) (
v2 v3 v4 h h h ) ( a ) (
a )
where a denotes an element of the original matrix A, h
denotes a modified element of the upper Hessenberg matrix H,
and vi denotes an element of the vector defining H(i).